Chapter 3.1

Similarity Symmetry
Groups of
Rosettes S20

The idea of similarity symmetry and the possibility for its exact
mathematical treatment was introduced in the monograph by H. Weyl
(1952), who defines two similarity transformations of the plane
E2: a central dilatation (or simply, dilatation) and dilative
rotation, with the restriction for the dilatation coefficient
k > 0, and he establishes the connection between the
transformations mentioned and the corresponding space isometries
- a translation and twist, respectively. His analysis is based
on natural forms satisfying similarity symmetry (e.g., the
Nautilus shell, Figure 3.1.; the sunflower Heliantus
maximus, etc.). In considering a spiral tendency in nature Weyl
quotes certain older authors (e.g., Leonardo and Goethe), who
also studied these problems and also that of a phyllotaxis,
the connection between the way of growth of certain
plants and the Fibonnaci sequence, linked to a golden
section (H.S.M. Coxeter, 1953, 1969). The sequence
1,1,2,3,5,8,13¼ defined by the
recursion formula: f1 = 1,
f2 = 1,
fn+fn+1 = fn+2,
nÎN, is called the Fibonnaci
sequence. A golden section ("aurea sectio" or "de
divina proportione", according to L. Paccioli) is the division
of a line segment so that the ratio of the larger part to the
smaller is equal to the ratio of the whole segment to the larger
part, i.e. its division in the ratio t:1, where t is
the positive root of the quadratic equation
t2+t+1 = 0,
t = (Ö5+1)/2
» 1,618033989...

Figure 3.1

Cross-section of a Nautilus shell.

The next step in the development of the theory of
similarity symmetry in the plane E2 was a contribution by
A.V. Shubnikov (1960). He described all the similarity
transformations of the plane E2: central dilatation K,
dilative rotation L and dilative reflection M and the
symmetry groups derived by one of the transformations mentioned
and by isometries having the same invariant point - rotations
and reflections. Shubnikov derived six types of discrete
similarity symmetry groups of rosettes S20: CnK,
CnL, CnM, DnK, DnL,
DnM, denoted by Shubnikov nK, nL, nM, nmK,
nmL, nmM respectively. Since the types DnM
(nmM) and DnL (nmL) coincide, there
are, in fact, five types of the discrete similarity symmetry
groups of rosettes S20: CnK (nK),
CnM (nM), CnL (nL), DnK (
nmK), Dn L (nmL) and two types of the visually
presentable continuous similarity symmetry groups of rosettes
S20: D¥ K (¥K) and
CnL1 (nL1). The term "type of similarity symmetry groups of
rosettes" and the corresponding type symbol denote all the
similarity symmetry groups defined by this symbol, that can be
obtained by different combinations of parameters defining them.
For example, by the symbol CnK (nK) are denoted
all the corresponding similarity symmetry groups which can be
obtained for different values of n
(nÎN) and k
(where K = K(k)).

Form of the fundamental region:
bounded, allows changes of the shape of
boundaries that
do not belong to reflection lines,
so symmetry groups of
the types CnK (nK),
CnL (nL),
CnM (nM)
allow changes
of the shape of all the boundaries, while symmetry groups
of the types DnK (nmK),
DnL (nmL) allow only
changes
of the shape of boundaries that do not belong to reflection
lines.

Further analysis on similarity symmetry groups was
undertaken by E.I. Galyarski and A.M. Zamorzaev (1963). Besides
giving the precise definitions of the similarity transformations
K, L, M, they used the adequate names for these
transformations, comparing them, respectively, with the
corresponding isometries of the space E3 - translation,
twist and glide reflection. They also successfully established
the isomorphism between the similarity symmetry groups of
rosettes S20 and the corresponding symmetry groups of
oriented, polar rods G31. In this way, consideration of the
similarity symmetry groups of rosettes S20 and their
generalizations is reduced to the consideration of the
corresponding, far better known symmetry, antisymmetry and
color-symmetry groups of polar, oriented rods G31. The
principle of crystallographic restriction (n = 1, 2, 3, 4, 6)
is followed by E.I. Galyarski and A.M. Zamorzaev.

In the work by E.I. Galyarski and A.M. Zamorzaev (1963),
there is no the restriction for the dilatation coefficient
k > 0, used by H. Weyl (1952).
This restriction does not result
in any loss of generality, but only in the somewhat different
classification of the similarity symmetry groups of rosettes
S20.

There is also the problem that for every particular
similarity symmetry group of rosettes S20,
its corresponding type is not always uniquely defined. Namely, under certain
conditions, the same symmetry group can be included in two
different types. Such a case is, e.g., that symmetry groups of the
type CnK (nK), because of the relationship
K(k) = L(k,0), also belong to the type CnL (nL).
If we accept the condition K = K(k) = L(k,0) = L0, then there also
exists the subtype DnL0 (nmL0), but symmetry
groups of the subtype mentioned are not included in the type
DnL2n (nmL2n).
If we accept the criterion of
subordination, which means, if we consider symmetry groups
existing in two different types within the larger type, certain
types would not exist at all. For example, all the symmetry
groups of the type CnK (nK) would be included in
the type CnL (nL), so that the type CnK
(nK) would not exist at all, and so on. A similar problem
may occur with the same similarity symmetry group that can
be defined by different sets of parameters n, k,
q... To consequently solve that
problem, it is necessary
to accept the common criterion of maximal symmetry. Such an
overlapping of different types of the similarity symmetry groups
of rosettes S20 is possible to avoid by accepting Weyl's
condition k > 0 for all
the similarity symmetry groups of rosettes S20
and the condition 0 < |q| < p/n
for symmetry groups of the type CnL (nL).